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Superselection Sectors in Quantum Spin Systems∗

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Superselection Sectors in Quantum Spin Systems∗ Superselection sectors in quantum spin systems∗ Pieter Naaijkens Leibniz Universit¨atHannover June 19, 2014 Abstract In certain quantum mechanical systems one can build superpositions of states whose relative phase is not observable. This is related to super- selection sectors: the algebra of observables in such a situation acts as a direct sum of irreducible representations on a Hilbert space. Physically, this implies that there are certain global quantities that one cannot change with local operations, for example the total charge of the system. Here I will discuss how superselection sectors arise in quantum spin systems, and how one can deal with them mathematically. As an example we apply some of these ideas to Kitaev's toric code model, to show how the analysis of the superselection sectors can be used to get a complete un- derstanding of the "excitations" or "charges" in this model. In particular one can show that these excitations are so-called anyons. These notes introduce the concept of superselection sectors of quantum spin systems, and discuss an application to the analysis of charges in Kitaev's toric code. The aim is to communicate the main ideas behind these topics: most proofs will be omitted or only sketched. The interested reader can find the technical details in the references. A basic knowledge of the mathematical theory of quantum spin systems is assumed, such as given in the lectures by Bruno Nachtergaele during this conference. Other references are, among others, [3, 4, 15, 12]. Parts of these lecture notes are largely taken from [12]. 1 Superselection sectors We will say that two representations π and ρ of a C∗-algebra are unitary equiv- alent (or simply equivalent) if there is a unitary operator U : Hπ !Hρ between the corresponding Hilbert spaces and in addition we have that Uπ(A)U ∗ = ρ(A) ∼ ∗ for all A 2 A. If this is the case, we also write π = ρ. In general, a C -algebra has many inequivalent representations. Here we discuss some of the consequences for quantum mechanical systems of the existence of inequivalent representations. In the next section we will consider an example of a system with inequivalent representations. This happens only for systems with infinitely many degrees of freedom. For finite systems the situation is different. For example, von Neu- mann [16] showed that there is only one irreducible representation of the canon- ical commutation relations [P; Q] = i~ (up to unitary equivalence). The same is ∗Lecture notes for a lecture given at NSF/CBMS Conference on Quantum Spin Systems. 1 true for a spin-1/2 system. If we consider the algebra generated by Sx;Sy;Sz, 2 2 2 3 satisfying [Si;Sj] = i"ijkSk and Sx + Sy + Sz = 4 I, each irreducible representa- tion of these relations is unitary equivalent to the representation generated by the Pauli matrices [18]. A similar results is true for a finite number of copies of such systems. We will see that the existence of inequivalent representations has consequences for the superposition principle. If π : A ! B(H) is a representation of a C∗-algebra, there is an easy way to obtain different states on A. Take any vector 2 H of norm one. Then the assignment A 7! h ; π(A) i defines a state. Such states are called vector states for the representation π. Note that by the GNS construction it is clear that any state can be realised as a vector state in some representation. Consider two states !1 and !2 that are both vector states for the same representation π. Hence there are vectors 1 and 2 such that !i(A) = h i; π(A) ii for all A 2 A. 2 2 Consider now = α 1 + β 2 with α; β 2 C such that jαj + jβj = 1 and both α and β are non-zero. Then !(A) = h ; π(A) i again is a state. However, it may be the case that the resulting state is not pure (even if the !i are) and we have a mixture 2 2 !(A) = jαj !1(A) + jβj !2(A): (1.1) If this is the case for any representation π in which !1 and !2 are vector states, we say that the two states are not superposable or not coherent. This situation was first analysed by Wick, Wightman and Wigner [17]. Theorem 1.1 ([2, Thm 6.1]). Let !1 and !2 be pure states. Then they are not superposable if and only if the corresponding GNS representations π!1 and π!2 are inequivalent. Proof. Consider a representation π such that !1 and !2 are vector states in this representation. Write i 2 H for the corresponding vectors. Then we can consider the subspaces Hi of H, defined as the closure of π(A) i. The projections on these subspaces will be denoted by Pi. Note that i is, by definition, cyclic for the representation π(A) restricted to Hi. Let us write πi for these restricted representations. But since the vectors i implement the state, it follows that the representation πi must be (unitary H !H equivalent to) the GNS representations π!i . Let U : 1 2 be a bounded linear map such that Uπ1(A) = π2(A)U for all A 2 A. By first taking adjoints, ∗ ∗ we then see that U Uπ1(A) = π1(A)U U. By irreducibility of π1 it follows that U ∗U = λI for some λ 2 C. In fact, λ must be real since U ∗U is self- adjoint. A similar argument holds for UU ∗. Hence by rescaling we can choose U to be unitary, unless U ∗U = 0. Hence non-zero maps U intertwining the representations only exist if π1 and π2 are unitarily equivalent. 0 To get back to the original setting, let T 2 π(A) . Then P2TP1 can be identified with a map U : H1 !H2 such that Uπ1(A) = π2(A)U. Conversely, 0 any such map can be extended to an operator in P2π(A) P1. Hence 0 P2π(A) P1 = f0g if and only if π1 and π2 are not unitarily equivalent. 0 Since I 2 π(A) it is clear that P1P2 = P2P1 = 0 if π1 and π2 are not equivalent. This implies that H1 and H2 are orthogonal subspaces of H. Hence h 2; π(A) 1i = 0 = h 1; π(A)i 2i: 2 2 2 Consequently, if = α 1 + β 2 with jαj + jβj = 1, then 2 2 !(A) := h ; π(A) i = jαj !1(A) + jβj !2(A): Hence !1 and !2 are not superposable. Conversely, suppose that π!1 and π!2 are unitarily equivalent. Then there 0 must be some unitary U in π(A) such that P2UP1 =6 0. This is only possible if there are vectors 'i 2 Hi such that h'2;U'1i= 6 0. Since π(A) 1 is dense in H1 (and similarly for H2), there must be A1;A2 2 A such that hπ(A1) 2; Uπ(A2) 1i 6= 0: (1.2) Set ' = U 1. Since U commutes with π(A) for every A, it follows that h'; π(A)'i = !1(A): Now consider the vector = α' + β 2. This induces a state !, and we find 2 2 !(A) − jαj !1(A) − jβj !2(A) = αβhU 1; π(A) 2i + αβh 2; π(A)U 1i: (1.3) ∗ Consider then A = A2A1, where A1 and A2 are as above. Then the right hand side of equation (1.3) becomes 2 Re(αβhπ(A1) 2; Uπ(A2) 1i): By choosing a suitable multiple if A we can make the right hand side non-zero, because of equation (1.2). It follows that equation (1.1) does not hold. This result shows that as soon as a C∗-algebra has inequivalent representa- tions, there are states that are not coherent. That is, there are pure states !1 and !2 such that a superposition of those states is never pure. The proof also makes clear that if we have vector states corresponding to inequivalent repre- sentations, there can never be a transition from one state to the other, not even by applying any operation available in A, because h 1; π(A) 2i is zero. Such a rule that forbids such transitions is called a superselection rule. There are many different (but strongly related) notions of a superselection rule around, see for example [8] for a discussion. As an example of irreducible representations we consider a spin-1/2 chain on the line, that is, Γ = Z. We define a vector jfsngi by specifying the spin in the z-direction for each site n. In particular, we set + to be the state where each − sn = +1, and 1 the state with each sn = −1. That is, the states are those with all spins in the up direction, and all in the down direction. This induces states ! on the quasilocal algebra A. Using the GNS representation we can then get representations (π; H; ). The representation works in the way one would expect, by acting with the Pauli matrices on the individual sites. It is however a good idea to look a bit more closely at what the Hilbert spaces are. The cyclic vectors are just the states defined above. According to the GNS construction, the Hilbert space is then generated by acting with local observables on this state. But a local observable can only flip a finite number of spins. This implies that the Hilbert space H+ is spanned by vectors jfsngi with only finitely many sn = −1.
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